Problem 5.27: Centrifugal pump/turbine
The sketch shows a centrifugal turbine or pump with an impeller in a flat-sided casing of width b. A liquid of density r flows through the pump at a volume flow rate Q as shown at left. The figure at the right shows the blades as seen from the direction of the flow arrow in the left hand figure (you would see this if the casing on the inlet side were removed). The blades are mounted on a thin disc that is attached to the shaft.
Assuming that (i) the liquid enters with a purely axial velocity and (ii) exits at radius r2 with a velocity that follows the (moving) blade surfaces, and (iii) neglecting frictional effects,
(a) derive an expression for the clockwise external torque (see the figure at right), TS, exerted by the fluid on the shaft at a clockwise angular speed w. Express it in terms of Q, r, b, r1, r2, b1,b2 and w.
(b) What is the system's run-away speed w0, that is, the (clockwise) speed of the shaft running unrestrained? What is the torque Ts0 if the shaft is held at zero rotation?